Amemiya Norm Equals Orlicz Norm in Musielak-Orlicz Spaces

Let （Ω,μ） be a a-finite measure space and Φ ： Ω × [0,∞） → [0, ∞] be a Musielak-Orlicz function. Denote by L^Φ（Ω） the Musielak-Orlicz space generated by Φ. We prove that the Amemiya norm equals the Orlicz norm in L^Φ（Ω）.     

Orlicz spaces without extreme points

Orlicz function and sequence spaces unit balls of which have no extreme points are completely characterized for both (the Orlicz and the Luxemburg) norms. Their subspaces of order continuous elements, with the norms induced from the whole Orlicz spaces without extreme points in their unit balls are also characterized. The well-known spaces L₁ and c₀ with unit balls without extreme points are covered by our results. Moreover, a new example of a Banach space without extreme points in its unit ball is given (see Example 1). This is the subspace a(L₁+L_∞) of order continuous elements of the space L₁+L_∞ equipped with the norm whenever 0<a<∞ and μ(T)>1/a.     

On a class of Köthe sequence spaces with normal structure

In this note, we investigate the generalized modulus of convexity δ ( λ ) and the generalized modulus smoothness ρ ( λ ) . We obtain some estimates of these moduli for X = lp . We obtain inequalities between WCS coefficient of a K¨othe sequence space X and δ ( λ ) X . We show that, for a wide class of K¨othe sequence spaces X, if for some ε∈ (0, 9 10 ] holds δ X (ε) > 1 3 1 √ 3 2 ε, then X has normal structure.     

Analytic norms in Orlicz spaces

It is shown that an Orlicz sequence space admits an equivalent analytic renorming if and only if it is either isomorphic to or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent -Fréchet norm, but no equivalent analytic norm.

     

Weakly convergent sequence coefficient in Köthe sequence spaces

In this paper, we have discussed the weakly convergent sequence coefficient in Köthe sequence spaces with as their boundedly complete basis. Using those results, we can easily calculate the weakly convergent sequence coefficient in Orlicz sequence spaces.

     

Uniform rotundity of Orlicz function spaces equipped with the ‐Amemiya norm

Necessary and sufficient conditions for uniform rotundity of Orlicz function spaces equipped with the p‐Amemiya norm are presented. The obtained results unify, complete and widen the characterization of uniform rotundity of Orlicz spaces. In the case of the ∞‐Amemiya (i.e. the Luxemburg) norm or the 1‐Amemiya (i.e. the Orlicz) norm, these results were known earlier. Some connections with the fixed point theory and the best approximation theory are presented.     

Generalized difference sequence spaces on seminormed space defined by Orlicz functions

In this paper we define the sequence space ℓ _M (Δ ^m , p, q, s) on a seminormed complex linear space by using an Orlicz function. We study its different algebraic and topological properties like solidness, symmetricity, monotonicity, convergence free etc. We prove some inclusion relations involving ℓ _M (Δ ^m , p, q, s).      

On maximal functions in Orlicz spaces

Let and be the functions having the representations and , where is a positive continuous function such that and is quasi-increasing. Then the maximal function is a function in Orlicz space for all if and only if there exists a positive constant such that for all .

     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

WCS Coefficients in Orlicz Sequence Spaces

We introduce some practical calculation of the weakly convergent sequence coefficients of Orlicz sequence spaces equipped with Luxemburg norm and Orlicz norm. For the N-function (u) of which the index function is monotonuous, the exact value WCS(l⁽⁾) of Orlicz sequence space l⁽⁾ with Luxemburg norm is available, i.e. WCS(l⁽⁾) = or WCS(l) of l with Orlicz norm has the exact value or estimation

Weak Uniform Normal Structure and Fixed Points of Asymptotically Regular Semigroups

Let X be a Banach space with a weak uniform normal structure and C a non–empty convexweakly compact subset of X. Under some suitable restriction, we prove that every asymptoticallyregular semigroup T = {T(t) : t ∈¸ S} of selfmappings on C satisfying

${\mathop {\lim \inf }\limits_{S \mathrel\backepsilon t \to \infty } }{\left| {{\left\| {T(t)} \right\|}} \right|} < {\text{WCS}}(X)$

has a common fixed point, where WCS(X) is the weakly convergent sequence coefficient of X, and\({\left| {{\left\| {T(t)} \right\|}} \right|}\) is the exact Lipschitz constant of T(t).     

On the Duality Mapping Sets in Orlicz Sequence Spaces

The criteria for the weak compactness of duality mapping sets J(x) = {f∈X* : <f, x> = ∥f∥² = ∥x∥²} in Orlicz sequence spaces endowed either with the Luxemburg norm or with the Orlicz norm are obtained. Supported by the National Natural Science Foundation of China, Grants 19901007 and 19871020     

Orlicz序列空间的对偶映射

本文研究Orlicz序列空间的对偶映射,给出对偶映象集为弱紧集的充分必要 条件.作为推论,得到Orlicz序列空间的弱接近光滑性的充要条件.     

Local rotundity structure of Cesàro-Orlicz sequence spaces

Some criteria for extreme points and strong U-points in Cesàro-Orlicz spaces are given. In consequence we find a Cesàro-Orlicz sequence space different from c₀ which has no extreme points. Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. Various examples presented in this paper show that there are some differences between criteria for extreme points and strong U-points in Orlicz spaces and in Cesàro-Orlicz spaces. We also show that the uniqueness of the local best approximation needs the notion of SU-point, that is, the notion of the extreme point is not strong enough here.     

Normal structure and Pythagorean approach in Banach spaces

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>X$ be a real Banach space and $S(X) = \{x \in X: \|x\| = 1\}$ be the unit sphere of $X$. The parameters $E_{\epsilon}(X)=\sup\{\alpha_{\epsilon}(x): x \in S(X)\}$, $e_{\epsilon}(X)=\inf\{\alpha_{\epsilon}(x): x \in S(X)\}$, $F_{\epsilon}(X)=\sup\{\beta_{\epsilon}(x): x \in S(X)\}$, and $f_{\epsilon}(X)=\inf\{\beta_{\epsilon}(x): x \in S(X)\}$, where $\alpha_{\epsilon}(x) = \sup\{\| x + \epsilon y \|^{2}+ \| x - \epsilon y \|^{2}: y \in S(X)\}$ and $\beta_{\epsilon}(x) = \inf\{\| x + \epsilon y \|^{2}+ \| x - \epsilon y \|^{2}: y \in S(X)\}$, are defined and studied. The main result is that a Banach space $X$ with $E_{\epsilon}(X) < 2 + 2\epsilon +\frac{1}{2}\epsilon^{2}$ for some $0\leq \epsilon \leq 1$ has uniform normal structure.     

ON A GENERALIZED MODULUS OF CONVEXITY AND UNIFORM NORMAL STRUCTURE

In this article, the authors study a generalized modulus of convexity, δ(α)(∈).Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ∈, 0 ≤∈≤1, such that δ(α)(1 ∈) ＞ (1 - α)∈.     

赋Orlicz范数下Orlicz-Lorentz序列空间的一致正规结构和一致非方性

Orlicz-Lorent空间作为经典Orlicz空间的推广,为调和分析及微分方程提供了合理的空间框架,而一致正规结构和一致非方性在不动点理论领域有着重要的应用.本文给出了赋Orlicz范数下Orlicz-Lorentz序列空间具有一致正规结构和一致非方性的充要条件.     

赋Orlicz范数的Musielak-Orlicz序列空间的Gateaux可微点与Frechet可微点

该文给出赋0rlicz范数的Musielak-Orlicz序列空间中Gateaux可微点(光滑点)与Frechet可微点(强光滑点)的判定准则．在此基础上推出了该空间具有光滑性或强光滑性的充分必要条件．     

Banach space properties sufficient for normal structure

In this paper we establish lower bounds for the weakly convergent sequence coefficient WCS(X) of a Banach space X, in terms of some well known moduli and coefficients. By mean of these bounds we identify several properties, of geometrical nature, which imply normal structure. We show that these properties are strictly more general than other previously known sufficient conditions for normal structure.     

Complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm

In this paper, two equivalent definitions of complex strongly extreme points in general complex Banach spaces are shown. It is proved that for any Orlicz sequence space equipped with the p-Amemiya norm (1?p<∞, p is odd), complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in Orlicz sequence spaces equipped with the p-Amemiya norm are given. Criteria for complex mid-point locally uniform rotundity and complex rotundity of Orlicz sequence spaces equipped with the p-Amemiya norm are also deduced.